Real, the

These coupling matrix elements are scalars due to the presence of the scalar Laplacian V. in Eq. (25). These elements are, in general, complex but if we require the to be real they become real. The matrix unlike its... 

Requiring l/f (r qx) to be real, the matrix W (Rx) becomes real and skew-symmetiic (just like its adiabatic counterpart) with diagonal elements equal to zero. Similarly, W (Rx) is an n X u diabatic second-derivative coupling matrix with elements defined by... 

Dry gas seals operate under extremely tight toleranees, whieh demand that speeial eare be taken in the design of the gas seal environment, and in the operation of the eompressor and gas seal system. While the threat of seal degradation and redueed seal life due to outside influenees is real, the detrimental effeets of these faetors ean be minimized. 

Suppose we have a wavefunction P, which for the sake of argument I will assume to be real. The expectation value of the energy is... 

Rule 1 Individual resonance forms are imaginary, not real. The real structure is a composite, or resonance hybrid, of the different forms. Species such as the acetate ion and benzene are no different from any other. They have single, unchanging structures, and they do not switch back and forth between resonance forms. The only difference between these and other substances is in the way they must be represented on paper. 

If A is real, the occurrence of conjugate complex roots is, of course, quite possible. The simplest of all methods is to take an arbitrary initial vector v0 and to form the sequence of iterates... 

The requirement that x x = xlixll is the mathematical statement that light propagates with the same speed with respect to both 0 and O, if they are equivalent observers. In (9-8) the transformation coefficients A"v are all real. The condition (9-10) requires that... 

Because of the hermitioity of p, the coefficients a and g are real. The requirement that Tr p — 1 implies that a = 1, while the equation p2 — p requires that 1 hence... 

The vinylidene group, the rro s-CPh=C- group, and the methylene group, all of which have one atom intervening between the substituent and the reaction site, all show Ji values of about 4. The cyclopropylidene group is an exception to this observation, with Ji of 6.38. As there were only four points in the set from which a was obtained, this deviation may not be real. The C—CPh=C group has a somewhat low value for 7j of 3.39. 

Since A is hermitian and a2 is real, the left-hand side may be written as... 

Since has been already constrained to be hermitian, it is legitimate to assume, withoutany loss of generality that is always diagonalizable into, say, , by a unitary transformation of the basis elements [10], The diagonal elements of , then called its eigenvalues, are real. The rank constraint on P (which is basis independent) further reduces the number of non-zero eigenvalues toN. Let % (i = l,. .., N), be these non-zero eigenvalues. 

The ensemble of rational and irrational numbers am called real numbers. Clearly, the sum, difference and product of two real numbers is real. The division of two ted numbers is defined in all cases but one - division by zero. Your computer will spit out an error message if you try to divide by zero ... 

As indicated above, if the components of a vector X in n-dimensional space are real, the vector can be written as a column matrix with n rows. Similarly, a second vector Y in the same space can be written as a column matrix of the... 

We have selected several unblended CH lines located in the near-UV between 3145-3190 A modified their oscillator strengths (gf-values) by fitting the solar high-resolution spectrum and assuming solar abundance of carbon 8.56 from Anders and Grevesse (1989). These lines are measurable in dwarfs down to the metallicities —3. Our results are shown in the Fig. 1. We confirm the metallicity dependence of the C/O ratio (Tomkin et al. 1992, Akerman et al. 2004). On the other hand, our plot for C/O shows a steep rise at [0/H]< —1. It is not clear if this effect is real. The work is in progress to address this issue using other abundance indicators such as CH 4300 A and better quality spectra. 

The expression obtained by this procedure is a real function. Thus, in many instances we will use the product ip ip instead of ip2, although if ip is real, the two are equivalent. 

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